We study affine permutation diagrams and their labellings with positive integers. Balanced labellings of a Rothe diagram of a finite permutation were defined by FominGreene-Reiner-Shimozono, and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column...

It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1, 2, . . . , n without repetition. These labellings are called Sn EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a n...

A graceful labeling of a graph G of size n is an injective assignment of integers from {0, 1,..., n} to the vertices of G, such that when each edge of G has assigned a weight, given by the absolute dierence of the labels of its end vertices, the set of weights is {1, 2,..., n}. If a graceful labeling f of a bipartite graph G assigns the smaller labels to one of the two stable sets of G, then f ...

A digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from {0, 1, ..., q} to each vertex v. The vertex values, in turn, induce a value g(u, v) on each arc (u, v) where g(u, v) = (g(v)− g(u))(mod q + 1). If the arc values are all distinct then the labeling is called a graceful labeling of a digraph. In this paper, we prove a general result on graceful digra...

In graph theory, a graceful labeling of a graph G = (V, E) with n vertices and m edges is a labeling of its vertices with distinct integers between 0 and m inclusive, such that each edge is uniquely identified by the absolute difference between its endpoints. In this paper, the well-known graceful labeling problem of graphs is represented as an optimization problem, and an algorithm based on An...

The first problem we deal with in this dissertation concerns long induced cycles in the d-dimensional cube, i.e. the graph on d-tuples of binary digits with pairs of vertices differing in exactly one coordinate as edges. Such cycles were introduced by Kautz as a type of error-checking codes (called the snake-in-the-box codes) for a certain analogue-to-digital conversion system. We give a simple...

It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1, 2, . . . , n without repetition. These labellings are called Sn EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a n...

A graceful labelling of a graph with n edges is a vertex labelling where the induced set of edge weights is {1, . . . , n}. A near graceful labelling is almost the same, the difference being that the edge weights are {1, 2, . . . , n − 1, n + 1}. In both cases, the weight of an edge is the absolute difference between its two vertex labels. Rosa [8] in 1988 conjectured that all triangular cacti ...